Finite Group Databasebeta

← #26226852

Subgroups — #26226852

Order 12032.

830 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689178C211yesmaximal, characteristic
2#20245794C211yesmaximal, characteristic
3#20245793C211yesmaximal, characteristic
4#20244150C212yesmaximal, complemented
5#20244145C212yesmaximal, complemented
6#20243590C212yesmaximal
7#20243588C212yesmaximal
8#20243557C211yesmaximal, characteristic, complemented
9#20243556C211yesmaximal, characteristic, complemented
10#20243523C211yesmaximal, characteristic, complemented
11C47×#1836C211yesmaximal, characteristic, complemented
12C2×#13093982—21no—
13C2×#13093982C2211yescharacteristic
14C2×#13093982C2211yescharacteristic
15C2×C42×D47C2211yescharacteristic
16#15689178C418yescomplemented
17C2×#13093958C2212yes—
18C2×#13093957C2212yes—
19C2×#13093956C2212yes—
20C2×#13093955C2212yes—
21C2×Dic47⋊C8C2212yes—
22C2×(C2×D47)⋊C8C2212yes—
23C2×#13093935C2211yescharacteristic
24#15688993C2211yescharacteristic
25#15688992C2211yescharacteristic
26#15688992C2211yescharacteristic
27#15688989C2211yescharacteristic
28#15688988—21no—
29#15688982—21no—
30#15688982—21no—
31#15688980—21no—
32#15688980C2212yes—
33C4⋊C4×Dic47—21no—
34C4⋊C4×Dic47C2212yes—
35#15688974—21no—
36#15688974—21no—
37#15688885C2212yes—
38#15688885C2212yes—
39C2×#97×C47C2211yescharacteristic
40C47×#384C2211yescharacteristic
41C47×#382C2211yescharacteristic
42C47×#340C2212yes—
43C47×#340C2212yes—
44C22×C4×D47D411yescharacteristic
45C22×C4×D47D411yescharacteristic
46C22×C4×D47C2311yescharacteristic
47C22×C47⋊C8C2312yes—
48#13093982—42no—
49#13093982—42no—
50#13093982—44no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47C2312yes—
54C2×C4⋊Dic47C2311yescharacteristic
55C2×C4⋊Dic47C2311yescharacteristic
56C2×Dic47⋊C4—21no—
57C2×Dic47⋊C4—21no—
58C2×Dic47⋊C4—21no—
59C2×Dic47⋊C4—21no—
60C2×(C2×D47)⋊C4—22no—
61C2×(C2×D47)⋊C4—21no—
62C2×(C2×D47)⋊C4—21no—
63C2×C4×Dic47—21no—
64C2×C4×Dic47—21no—
65C2×C4×Dic47—22no—
66C2×C4×Dic47D411yescharacteristic
67C2×C4×Dic47D411yescharacteristic
68C2×C4×Dic47C2311yescharacteristic
69C42×D47C2×C414yes—
70C42×D47Q814yes—
71#13093958—42no—
72#13093957—42no—
73#13093956—42no—
74#13093955—42no—
75Dic47⋊C8C2×C418yes—
76(C2×D47)⋊C8C2×C418yes—
77#13093935C2×C414yes—
78C47⋊#95—21no—
79C47⋊#95—21no—
80C47⋊#95—22no—
81C47⋊#95—22no—
82C22×C8×C47C2312yes—
83C2×C4⋊C4×C47—21no—
84C2×C4⋊C4×C47—21no—
85C2×C4⋊C4×C47C2312yes—
86C2×C4⋊C4×C47C2311yescharacteristic
87C2×C4⋊C4×C47C2311yescharacteristic
88C2×C42×C47C2311yescharacteristic
89#97×C47C2×C414yes—
90#95×C47—21no—
91#95×C47—21no—
92C23×D47C2×D411yescharacteristic
93C22×C4×C47—22no—
94C22×C4×C47—21no—
95C22×C4×C47—21no—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47C2411yescharacteristic
99C22×Dic47—21no—
100C22×Dic47—21no—
101C22×Dic47—22no—
102C22×Dic47C2×D411yescharacteristic
103C22×Dic47C2×D411yescharacteristic
104C22×Dic47C2×D411yescharacteristic
105C2×C4×D47—22no—
106C2×C4×D47—22no—
107C2×C4×D47—22no—
108C2×C4×D47—22no—
109C2×C4×D47—41no—
110C2×C4×D47—41no—
111C2×C4×D47C4⋊C414yes—
112C2×C4×D47C4⋊C414yes—
113C2×C4×D47C22×C412yes—
114C2×C4×D47C2×Q812yes—
115C2×C8×C47—22no—
116C2×C8×C47—22no—
117C2×C8×C47C22×C414yes—
118C4⋊C4×C47—22no—
119C4⋊C4×C47—22no—
120C4⋊C4×C47—24no—
121C4⋊C4×C47—42no—
122C4⋊C4×C47—42no—
123C42×C47C22×C412yes—
124C42×C47C2×Q812yes—
125(C2×D47)⋊C4—42no—
126(C2×D47)⋊C4—42no—
127(C2×D47)⋊C4—44no—
128C4⋊Dic47—22no—
129C4⋊Dic47—22no—
130C4⋊Dic47—22no—
131C4⋊Dic47—22no—
132C4⋊Dic47—42no—
133C4⋊Dic47—42no—
134Dic47⋊C4—42no—
135Dic47⋊C4—42no—
136Dic47⋊C4—42no—
137Dic47⋊C4—42no—
138C4×Dic47—42no—
139C4×Dic47—42no—
140C4×Dic47—44no—
141C4×Dic47C4⋊C414yes—
142C4×Dic47C4⋊C414yes—
143C4×Dic47C22×C412yes—
144C4×Dic47C2×Q812yes—
145C2×C47⋊C8—22no—
146C2×C47⋊C8—22no—
147C2×C47⋊C8C22×C414yes—
148C23×C47C22×D411yescharacteristic
149C22×D47—21no—
150C22×D47—21no—
151C22×D47—21no—
152C22×D47—21no—
153C22×D47—21no—
154C22×D47—21no—
155C22×D47C2×C4⋊C412yes—
156C2×C4×C47—21no—
157C2×C4×C47—21no—
158C2×C4×C47—21no—
159C2×C4×C47—21no—
160C2×C4×C47—41no—
161C2×C4×C47—41no—
162C2×C4×C47—42no—
163C2×C4×C47—42no—
164C2×C4×C47—41no—
165C2×C4×C47—41no—
166C2×C4×C47—41no—
167C2×C4×C47—41no—
168C2×C4×C47—42no—
169C2×C4×C47C2×C4⋊C412yes—
170C2×C4×C47C2×C4⋊C412yes—
171C2×C4×C47C23×C411yescharacteristic
172C2×C4×C47C22×Q811yescharacteristic
173C2×C4×C47H(5,𝔽2)12yes—
174C2×C4×C472−1+412yes—
175C2×Dic47—21no—
176C2×Dic47—21no—
177C2×Dic47—21no—
178C2×Dic47—21no—
179C2×Dic47—41no—
180C2×Dic47—41no—
181C2×Dic47—42no—
182C2×Dic47—42no—
183C2×Dic47—41no—
184C2×Dic47—41no—
185C2×Dic47—41no—
186C2×Dic47—41no—
187C2×Dic47—42no—
188C2×Dic47C2×C4⋊C412yes—
189C2×Dic47C2×C4⋊C412yes—
190C2×Dic47C2×C4⋊C412yes—
191C2×Dic47AGL(1,ℤ/8)12yes—
192C2×Dic47#13712yes—
193C4×D47—24no—
194C4×D47—24no—
195C4×D47—28no—
196C4×D47—44no—
197C8×C47—24no—
198C47⋊C8—24no—
199#21988—471nomaximal, Sylow
200C22×C47C22×C4⋊C411yescharacteristic
201C22×C47#53511yescharacteristic
202C22×C47#53611yescharacteristic
203C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
204C22×C47C2×#13711yescharacteristic
205C22×C47#57511yescharacteristic
206C22×C47#57511yescharacteristic
207C2×D47—22no—
208C2×D47—22no—
209C2×D47—22no—
210C2×D47—41no—
211C2×D47—41no—
212C2×D47—41no—
213C2×D47—41no—
214C4×C47—22no—
215C4×C47—22no—
216C4×C47—81no—
217C4×C47—81no—
218C4×C47—82no—
219C4×C47#52012yes—
220C4×C47#54412yes—
221Dic47—22no—
222Dic47—22no—
223Dic47—81no—
224Dic47—82no—
225Dic47—81no—
226Dic47#42814yes—
227C2×#432—471no—
228#2285—471no—
229#2285—471no—
230#1948—472no—
231#1947—472no—
232#1854—472no—
233#1854—472no—
234#1847—471no—
235#1847—471no—
236#1836—471no—
237#1836D4711yescharacteristic, complemented
238C2×C47#231711yescharacteristic
239C2×C47C2×#42811yescharacteristic
240C2×C47#291111yescharacteristic
241C2×C47#321411yescharacteristic
242C2×C47#321611yescharacteristic
243C2×C47#323511yescharacteristic
244C2×C47#323711yescharacteristic
245D47—42no—
246C2×#117—471no—
247C2×#117—471no—
248C2×#117—941no—
249C22×C42—471no—
250#432—478no—
251C2×#107—472no—
252C2×#107—472no—
253C2×#106—472no—
254C2×#106—472no—
255C2×C4⋊C8—472no—
256C2×C22⋊C8—472no—
257C2×#97—471no—
258C2×#97C2×D4711yescharacteristic
259#384—471no—
260#384—471no—
261#384—471no—
262#384C2×D4711yescharacteristic
263#382—471no—
264#382—941no—
265#382C2×D4711yescharacteristic
266#380—472no—
267#380—941no—
268#380—941no—
269#380—941no—
270C4×C4⋊C4—472no—
271C4×C4⋊C4—941no—
272C4×C4⋊C4—941no—
273C4×C4⋊C4—941no—
274#340—472no—
275#340—472no—
276#340C2×D4712yes—
277#340C2×D4712yes—
278C47#2198811yescharacteristic, complemented, Sylow
279C23×C4—471no—
280C23×C4—471no—
281C23×C4—471no—
282C22×C8—472no—
283C22×C8C22×D4712yes—
284#117—1882no—
285#117—1882no—
286#117—1884no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—471no—
290C2×C4⋊C4—471no—
291C2×C4⋊C4—472no—
292C2×C4⋊C4—941no—
293C2×C4⋊C4—941no—
294C2×C4⋊C4—941no—
295C2×C4⋊C4—941no—
296C2×C4⋊C4—941no—
297C2×C4⋊C4—941no—
298C2×C4⋊C4C22×D4711yescharacteristic
299C2×C4⋊C4C22×D4711yescharacteristic
300C2×C4⋊C4C22×D4712yes—
301C2×C22⋊C4—941no—
302C2×C22⋊C4—941no—
303C2×C22⋊C4—942no—
304C2×C42—471no—
305C2×C42—471no—
306C2×C42—471no—
307C2×C42—474no—
308C2×C42—474no—
309C2×C42—941no—
310C2×C42—941no—
311C2×C42—942no—
312C2×C42C22×D4711yescharacteristic
313#107—1882no—
314#107—1882no—
315#106—1882no—
316#106—1882no—
317C4⋊C8—478no—
318C22⋊C8—478no—
319#97—474no—
320#97C4×D4714yes—
321#95—21no—
322#95—21no—
323#95—942no—
324#95—942no—
325#95—941no—
326#95—941no—
327C24—471no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—22no—
331C22×C4—471no—
332C22×C4—471no—
333C22×C4—471no—
334C22×C4—474no—
335C22×C4—472no—
336C22×C4—472no—
337C22×C4—474no—
338C22×C4—942no—
339C22×C4—942no—
340C22×C4—942no—
341C22×C4—942no—
342C22×C4—942no—
343C22×C4—941no—
344C22×C4—941no—
345C22×C4—1881no—
346C22×C4—1881no—
347C22×C4D4×D4711yescharacteristic
348C22×C4D4×D4711yescharacteristic
349C22×C4C23×D4711yescharacteristic
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—474no—
353C2×C8—942no—
354C2×C8—942no—
355C2×C8C2×C4×D4714yes—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—942no—
362C4⋊C4—942no—
363C4⋊C4—942no—
364C4⋊C4—942no—
365C4⋊C4—1882no—
366C4⋊C4—1882no—
367C4⋊C4—1882no—
368C4⋊C4—1882no—
369C4⋊C4—1882no—
370C4⋊C4—1882no—
371C22⋊C4—1884no—
372C22⋊C4—1882no—
373C22⋊C4—1882no—
374C42—474no—
375C42—474no—
376C42—472no—
377C42—472no—
378C42—1884no—
379C42—1882no—
380C42—1882no—
381C42C2×C4×D4712yes—
382C42Q8×D4712yes—
383C23—472no—
384C23—941no—
385C23—941no—
386C23—941no—
387C23—941no—
388C23—941no—
389C23—941no—
390C23C2×D4×D4711yescharacteristic
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—472no—
405C2×C4—472no—
406C2×C4—472no—
407C2×C4—472no—
408C2×C4—472no—
409C2×C4—948no—
410C2×C4—941no—
411C2×C4—941no—
412C2×C4—941no—
413C2×C4—941no—
414C2×C4—944no—
415C2×C4—944no—
416C2×C4—1884no—
417C2×C4—1882no—
418C2×C4—1881no—
419C2×C4—1881no—
420C2×C4—1881no—
421C2×C4—1881no—
422C2×C4—1882no—
423C2×C4—1881no—
424C2×C4—1881no—
425C2×C4—1882no—
426C2×C4C4⋊C4×D4712yes—
427C2×C4C4⋊C4×D4712yes—
428C2×C4C22×C4×D4711yescharacteristic
429C2×C4C2×Q8×D4711yescharacteristic
430C2×C4#1309407012yes—
431C2×C4#1309407112yes—
432C8—24no—
433C8—944no—
434C22—942no—
435C22—942no—
436C22—942no—
437C22—1881no—
438C22—1881no—
439C22—1881no—
440C22—1881no—
441C22C2×C4⋊C4×D4711yescharacteristic
442C22#1568969111yescharacteristic
443C22#1568969311yescharacteristic
444C22AGL(1,ℤ/8)×D4711yescharacteristic
445C22#137×D4711yescharacteristic
446C22#1568991911yescharacteristic
447C22#1568992211yescharacteristic
448C4—22no—
449C4—22no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—474no—
454C4—942no—
455C4—942no—
456C4—3762no—
457C4—3761no—
458C4—3761no—
459C4#1568960312yes—
460C4#1568975312yes—
461C2—1882no—
462C2#2024601311yescharacteristic
463C2#428×D4711yescharacteristic
464C2#2025197511yescharacteristic
465C2#2025438411yescharacteristic
466C2#2025440011yescharacteristic
467C2#2025458111yescharacteristic
468C2#2025459711yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1836C2
socleC23×C47C22×D4
prime radicalC23×D47C2×D4
cosocleC22×C4×C47C24
hypercenter#1836D47
hypocenterC47#21988
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1836

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #20244150C2semidirect
5. #20244145C2semidirect
8. #20243557C2semidirect
9. #20243556C2semidirect
10. #20243523C2semidirect
11. C47×#1836C2semidirect
16. #15689178C4semidirect
237. #1836D47semidirect
278. C47#21988semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#156891782
#202457942
#202457932
#202441502
#202441452
#202435902
#202435882
#202435572
#202435562
#202435232
C47×#18362
#2198847